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Orthonormal bases of compactly supported wavelets
 Commun Pure Appl Math 41:906–966
, 1988
"... Abstract. Several variations are given on the construction of orthonormal bases of wavelets with compact support. They have, respectively, more symmetry, more regularity, or more vanishing moments for the scaling function than the examples constructed in Daubechies [Comm. Pure Appl. Math., 41 (1988) ..."
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Cited by 1570 (27 self)
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Abstract. Several variations are given on the construction of orthonormal bases of wavelets with compact support. They have, respectively, more symmetry, more regularity, or more vanishing moments for the scaling function than the examples constructed in Daubechies [Comm. Pure Appl. Math., 41 (1988), pp. 909996]. Key words, wavelets, orthonormal bases, regularity, symmetry AMS(MOS) subject classifications. 26A16, 26A18, 26A27, 39B12
Factoring wavelet transforms into lifting steps
 J. Fourier Anal. Appl
, 1998
"... ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This dec ..."
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Cited by 434 (7 self)
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ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is wellknown to algebraists (and expressed by the formula); it is also used in linear systems theory in the electrical engineering community. We present here a selfcontained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e, nonunitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a waveletlike transform that maps integers to integers. 1.
The Stationary Wavelet Transform and some Statistical Applications
, 1995
"... Wavelets are of wide potential use in statistical contexts. The basics of the discrete wavelet transform are reviewed using a filter notation that is useful subsequently in the paper. A `stationary wavelet transform', where the coefficient sequences are not decimated at each stage, is described. Two ..."
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Cited by 128 (19 self)
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Wavelets are of wide potential use in statistical contexts. The basics of the discrete wavelet transform are reviewed using a filter notation that is useful subsequently in the paper. A `stationary wavelet transform', where the coefficient sequences are not decimated at each stage, is described. Two different approaches to the construction of an inverse of the stationary wavelet transform are set out. The application of the stationary wavelet transform as an exploratory statistical method is discussed, together with its potential use in nonparametric regression. A method of local spectral density estimation is developed. This involves extensions to the wavelet context of standard time series ideas such as the periodogram and spectrum. The technique is illustrated by its application to data sets from astronomy and veterinary anatomy. 1 Introduction In this paper we discuss some aspects of wavelets with a particular view to their statistical application. In particular we shall be conce...
Perceptual Coding of Digital Audio
 Proceedings of the IEEE
, 2000
"... During the last decade, CDquality digital audio has essentially replaced analog audio. Emerging digital audio applications for network, wireless, and multimedia computing systems face a series of constraints such as reduced channel bandwidth, limited storage capacity, and low cost. These new applic ..."
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Cited by 105 (1 self)
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During the last decade, CDquality digital audio has essentially replaced analog audio. Emerging digital audio applications for network, wireless, and multimedia computing systems face a series of constraints such as reduced channel bandwidth, limited storage capacity, and low cost. These new applications have created a demand for highquality digital audio delivery at low bit rates. In response to this need, considerable research has been devoted to the development of algorithms for perceptually transparent coding of highfidelity (CDquality) digital audio. As a result, many algorithms have been proposed, and several have now become international and/or commercial product standards. This paper reviews algorithms for perceptually transparent coding of CDquality digital audio, including both research and standardization activities. The paper is organized as follows. First, psychoacoustic principles are described with the MPEG psychoacoustic signal analysis model 1 discussed in some detail. Next, filter bank design issues and algorithms are addressed, with a particular emphasis placed on the Modified Discrete Cosine Transform (MDCT), a perfect reconstruction (PR) cosinemodulated filter bank that has become of central importance in perceptual audio coding. Then, we review methodologies that achieve perceptually transparent coding of FM and CDquality audio signals, including algorithms that manipulate transform components, subband signal decompositions, sinusoidal signal components, and linear prediction (LP) parameters, as well as hybrid algorithms that make use of more than one signal model. These discussions concentrate on architectures and applications of
Modeling and estimation of multiresolution stochastic processes
 IEEE TRANS. ON INFORMATION THEORY
, 1992
"... An overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing. As described, a natural framework for developing such a theory ..."
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Cited by 94 (17 self)
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An overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing. As described, a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image. In particular, it will be seen how the wavelet transform directly suggests such a modeling paradigm. This perspective then leads directly to the investigation of several classes of dynamic models and related notions of “ multiscale stationarity ” in which scale plays the role of a timelike variable. Focus is primarily on the investigation of models on homogenous trees. In particular, the elements of a dynamic system theory on trees are described
Face Detection Using Quantized Skin Color Regions Merging and Wavelet Packet Analysis
, 1999
"... Detecting and recognizing human faces automatically in digital images strongly enhance contentbased video indexing systems. In this paper, a novel scheme for human faces detection in color images under nonconstrained scene conditions, such as the presence of a complex background and uncontrolled il ..."
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Cited by 84 (3 self)
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Detecting and recognizing human faces automatically in digital images strongly enhance contentbased video indexing systems. In this paper, a novel scheme for human faces detection in color images under nonconstrained scene conditions, such as the presence of a complex background and uncontrolled illumination, is presented. Color clustering and filtering using approximations of the YCbCr and HSV skin color subspaces are applied on the original image, providing quantized skin color regions. A merging stage is then iteratively performed on the set of homogeneous skin color regions in the color quantized image, in order to provide a set of potential face areas. Constraints related to shape and size of faces are applied, and face intensity texture is analyzed by performing a wavelet packet decomposition on each face area candidate in order to detect human faces. The wavelet coefficients of the band filtered images characterize the face texture and a set of simple statistical deviations is ...
The Discrete Wavelet Transform in S
 Journal of Computational and Graphical Statistics
, 1996
"... The theory of wavelets has recently undergone a period of rapid development. We introduce a software package called wavethresh that works within the statistical language S to perform one and twodimensional discrete wavelet transforms. The transforms and their inverses can be computed using any par ..."
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Cited by 81 (24 self)
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The theory of wavelets has recently undergone a period of rapid development. We introduce a software package called wavethresh that works within the statistical language S to perform one and twodimensional discrete wavelet transforms. The transforms and their inverses can be computed using any particular wavelet selected from a range of different families of wavelets. Pictures can be drawn of any of the one or twodimensional wavelets available in the package. The wavelet coefficients can be presented in a variety of ways to aid in the interpretation of data. The package's wavelet transform "engine" is written in C for speed and the objectorientated functionality of S makes wavethresh easy to use. We provide a tutorial introduction to wavelets and the wavethresh software. We also discuss how the software may be used to carry out nonlinear regression and image compression. In particular, thresholding of wavelet coefficients is a method for attempting to extract signal from noise and ...
Theory Of Regular MBand Wavelet Bases
 IEEE TRANS. ON SIGNAL PROCESSING
, 1993
"... This paper constructs Kregular Mband orthonormal wavelet bases. Kregularity of the wavelet basis is known to be useful in numerical analysis applications and in image coding using wavelet techniques. Several characterizations of Kregularity and their importance are described. An explicit formula ..."
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Cited by 79 (6 self)
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This paper constructs Kregular Mband orthonormal wavelet bases. Kregularity of the wavelet basis is known to be useful in numerical analysis applications and in image coding using wavelet techniques. Several characterizations of Kregularity and their importance are described. An explicit formula is obtained for all minimal length Mband scaling filters. A new statespace approach to constructing the wavelet filters from the scaling filters is also described. When Mband wavelets are constructed from unitary filter banks they give rise to wavelet tight frames in general (not orthonormal bases). Conditions on the scaling filter so that the wavelet bases obtained from it is orthonormal is also described.
Wavelet shrinkage using crossvalidation
, 1996
"... Wavelets are orthonormal basis functions with special properties that show potential in many areas of mathematics and statistics. This article concentrates on the estimation of functions and images from noisy data using wavelet shrinkage. A modified form of twofold crossvalidation is introduced to ..."
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Cited by 77 (13 self)
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Wavelets are orthonormal basis functions with special properties that show potential in many areas of mathematics and statistics. This article concentrates on the estimation of functions and images from noisy data using wavelet shrinkage. A modified form of twofold crossvalidation is introduced to choose a threshold for wavelet shrinkage estimators operating on data sets of length a power of two. The crossvalidation algorithm is then extended to data sets of any length and to multidimensional data sets. The algorithms are compared to established threshold choosers using simulation. An application to a real data set arising from anaesthesia is presented.
Wavelet Families Of Increasing Order In Arbitrary Dimensions
, 1997
"... . We build compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The resulting scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its ..."
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Cited by 46 (0 self)
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. We build compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The resulting scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its advantages: fast transform, inplace calculation, and integerto integer transforms. We show that two lifting steps suffice: predict and update. The predict step can be built using multivariate polynomial interpolation, while update is a multiple of the adjoint of predict. Submitted to IEEE Transactions on Image Processing Over the last decade several constructions of compactly supported wavelets have originated both from signal processing and mathematical analysis. In signal processing, critically sampled wavelet transforms are known as filter banks or subband transforms [32, 43, 54, 56]. In mathematical analysis, wavelets are defined as translates and dilates of one fixed function and ar...